It's not in SE1, but the area around Clerkenwell Road is the olde traditional watch repairers area of London (now limited to only near to the junction of St John Street) and there are a few independant watch repair shops still around there (they are very reasonably priced in my experience and I think it's nice to support the independant small businesses!)
The one that I've used before is @ 60-62 Clerkenwell Road - Called Watch Repair Service. You can see it on google streetview on the north side of the road.
I think there is another one on the other side too.
Actually, you might want to consider doing this yourself. (I did it myself just last night for the first time).
If it's just a leather strap, it probably comes with pins which can just compress easily and slide back into place.
You need patience, good lighting and eyesight, and a very small screwdriver (which I think Boots opticians sell for a couple of quid). It took me about 10 minutes. Be careful not to scratch the actual watch with the screwdriver.
Of course, that assumes that your watch strap is as simple as mine. If it's anything more complex, it's out of my depth! ;-) Perhaps an optician or shoe repair place could do it?
Just across the river in the City there is a watch repairers....across London Bridge and turn right into Eastcheap...about 20 meters and its there...they have fitted a few replacement batteries for me and are very quick and efficient...just a satisfied customer no other connection.
Go to PIccadilly Circus tube and walk around to the other side of the concourse from the barriers. There you will find a not so little man in what amounts to a cupboard. He'll do it for not many pennies at all. If you're really nice he might even do it for free as he did with me when he repaired one of the links on my watch bracelet, which had come loose.